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Birkhäuser

Minimal Surfaces and Functions of Bounded Variation

  • Book
  • © 1984

Overview

Authors:
  1. Enrico Giusti
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Part of the book series: Monographs in Mathematics (MMA, volume 80)

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Table of contents (17 chapters)

  1. Front Matter

    Pages i-xii
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  2. Parametric Minimal Surfaces

    1. Front Matter

      Pages 1-1
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    2. Functions of Bounded Variation and Caccioppoli Sets

      • Enrico Giusti
      Pages 3-29

    3. Traces of BV Functions

      • Enrico Giusti
      Pages 30-41

    4. The Reduced Boundary

      • Enrico Giusti
      Pages 42-51

    5. Regularity of the Reduced Boundary

      • Enrico Giusti
      Pages 52-62

    6. Some Inequalities

      • Enrico Giusti
      Pages 63-73

    7. Approximation of Minimal Sets (I)

      • Enrico Giusti
      Pages 74-84

    8. Approximation of Minimal Sets(II)

      • Enrico Giusti
      Pages 85-96

    9. Regularity of Minimal Surfaces

      • Enrico Giusti
      Pages 97-103

    10. Minimal Cones

      • Enrico Giusti
      Pages 104-114

    11. The First and Second Variation of the Area

      • Enrico Giusti
      Pages 115-127

    12. The Dimension of the Singular Set

      • Enrico Giusti
      Pages 128-134

  3. Non-Parametric Minimal Surfaces

    1. Front Matter

      Pages 135-135
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    2. Classical Solutions of the Minimal Surface Equation

      • Enrico Giusti
      Pages 137-150

    3. The a priori Estimate for the Gradient

      • Enrico Giusti
      Pages 151-159

    4. Direct Methods

      • Enrico Giusti
      Pages 160-171

    5. Boundary Regularity

      • Enrico Giusti
      Pages 172-181

    6. A Further Extension of the Notion of Non-Parametric Minimal Surface

      • Enrico Giusti
      Pages 182-200

    7. The Bernstein Problem

      • Enrico Giusti
      Pages 201-217
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Keywords

About this book

The problem of finding minimal surfaces, i. e. of finding the surface of least area among those bounded by a given curve, was one of the first considered after the foundation of the calculus of variations, and is one which received a satis­ factory solution only in recent years. Called the problem of Plateau, after the blind physicist who did beautiful experiments with soap films and bubbles, it has resisted the efforts of many mathematicians for more than a century. It was only in the thirties that a solution was given to the problem of Plateau in 3-dimensional Euclidean space, with the papers of Douglas [DJ] and Rado [R T1, 2]. The methods of Douglas and Rado were developed and extended in 3-dimensions by several authors, but none of the results was shown to hold even for minimal hypersurfaces in higher dimension, let alone surfaces of higher dimension and codimension. It was not until thirty years later that the problem of Plateau was successfully attacked in its full generality, by several authors using measure-theoretic methods; in particular see De Giorgi [DG1, 2, 4, 5], Reifenberg [RE], Federer and Fleming [FF] and Almgren [AF1, 2]. Federer and Fleming defined a k-dimensional surface in IR" as a k-current, i. e. a continuous linear functional on k-forms. Their method is treated in full detail in the splendid book of Federer [FH 1].

Bibliographic Information

  • Book Title: Minimal Surfaces and Functions of Bounded Variation

  • Authors: Enrico Giusti

  • Series Title: Monographs in Mathematics

  • DOI: https://doi.org/10.1007/978-1-4684-9486-0

  • Publisher: Birkhäuser Boston, MA

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Science+Business Media New York 1984

  • Softcover ISBN: 978-0-8176-3153-6Published: 01 January 1984

  • eBook ISBN: 978-1-4684-9486-0Published: 14 March 2013

  • Series ISSN: 1017-0480

  • Series E-ISSN: 2296-4886

  • Edition Number: 1

  • Number of Pages: XII, 240

  • Topics: Geometry

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